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Analysis (Kubo) Lab
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Research on the wave equation, which describes how waves propagate, from the viewpoints of perturbation and scattering.
AboutAbout the lab
Rich structure hidden in a simply shaped equation.
Prof. Kubo studies the wave equation, which describes how waves propagate.
The wave equation is a representative partial differential equation with a very simple form, yet the mathematical structure it contains is very rich.
Since graduate school, Prof. Kubo has continued to think about how solutions change when small changes (perturbations) are made to the nonlinearity of the equation, the domain, or the metric of the space.
Recently, Prof. Kubo has also become interested in systems of equations in mathematical physics that reduce to the wave equation, and in the noncommutative structure of differential equations.
Key terms
- Wave equation
- A representative partial differential equation that describes how waves travel through space.
- Perturbation
- Making a small change to the original equation or its conditions, in order to study how the behavior of solutions changes.
- Scattering theory
- A theory that studies the behavior of solutions as time goes to infinity by comparing them with the original solutions.
ResearchResearch themes
The lab examines how perturbations of the nonlinearity, the domain, and the metric of the space affect solutions of the wave equation, which describes wave propagation. Comparing the leading part of solutions of the unperturbed system with that of the perturbed solutions is the key to the research.
*Illustrative imagePerturbation Problems for the Wave Equation
The lab investigates how perturbations of the nonlinearity, perturbations of the domain, and perturbations of the metric of the space affect the original system. Cases in which the relationship between the leading parts of the two solutions is exactly balanced are especially interesting problems.
*Illustrative imageScattering Theory
This is a theory that compares the behavior of solutions as time goes to infinity, and it occupies a major place in the study of the wave equation. It also uses frameworks from functional analysis and real analysis, but the essence is how to derive the leading part of a solution, and in the end it comes down to demanding calculus computations.
*Illustrative imageSystems of Equations in Mathematical Physics
The lab is also interested in systems of equations arising in mathematical physics that reduce to the wave equation. The noncommutative structure of differential equations is another of its recent interests.
MembersFaculty
- ProfessorHideo Kubo
VoicesStudent voices
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “Surface waves” Roger McLassus / CC BY-SA 3.0 (cropped)
- Perturbation Problems for the Wave Equation: “A 3D Visualisation of 2 Source Interference in Ripple Tank” Psgs123xyz at English Wikipedia / CC0 (cropped)
- Scattering Theory: “Ripples dispersing sunlight into underwater rainbows in Brofjorden” W.carter / CC BY-SA 4.0 (cropped)
- Systems of Equations in Mathematical Physics: “Cubicharmonics 3840x2160” Daigokuz / CC BY-SA 3.0 (cropped)
- Related lab (Mathematical Sciences (Masaki) Lab): “Baltic Sea Wave (Cien Water)” Marek Nowocień / CC BY 2.5 (cropped)
- Related lab (Nakano Lab): “Hurricane Isabel from ISS” Image courtesy of Mike Trenchard, Earth Sciences & Image Analysis Laboratory, NASA Johnson Space Center. / Public domain (cropped)
- Related lab (Analysis Lab (Hamamuki)): “Real snowflake macro photo” Alexey Kljatov / CC BY-SA 4.0 (cropped)
- Related lab (Mathematical Sciences (Kuroda) Lab): “Newly fallen snow crystals accumulating on a window sill” W.carter / CC0 (cropped)
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